An explicit description of neighborhoods of the rank 2 boundary in the Satake Compactification of $\mathbf{A}_2$

My Motivation: I'm having a hard time following the description of the topology in the Satake Compactification of locally symmetric spaces. The group theory is something I'm finding a bit tricky to follow, so I'd prefer answers in terms of explicit matrix representations, though a concrete descriptions of cylindrical sets or something might be helpful too.

Background: Let $\mathbf{A}_2$ be the moduli space of dimension 2 principally polarized abelian varieties. If you prefer, just think of it as the Siegel upper half space of $2\times 2$ symmetric matrices $Z=X+iY$ with $Y$ positive definite, quotiented out by $\mathbf{SP}_4(\mathbf{Z})$. If one compactifies $\mathbf{A}_2$ according to the Satake compactification, there is a rank 1 boundary component and a rank 2 boundary component.

Main Question: What is the preimage- in Siegel upper half space - of a simple system of fundamental neighborhoods for the rank 2 boundary component of $\mathbf{A}_2$, in terms of an explicit description for the matrix co-ordinates.

Form of Answer that I'd like: Something about $Y$ as a quadratic form, involving its determinant, minimal vector length, etc...

I'd strongly prefer an answer entirely specific to this case, if possible, and not something for general locally symmetric spaces.

Additional question I'd be grateful to get the answer to: Same question but for the rank 1 boundary component.

Thanks!!

This answer is incomplete but I'll try to finish it later.

First note that you can't express the answer only using $Y$, even in the $g=1$ case. If the imaginary part of $\tau$ goes to $0$, you do not know anytihng about what $j(\tau)$ is doing.

I think you want to work with the induced quadratic form on pairs $(v,w) \in \mathbb Z^2 \times \mathbb Z^2 = \mathbb Z^4$ given by

$$(v_1,w_1) \cdot (v_2,w_2) = \operatorname{Re}\left( \overline{(v_1 + Z w_1)}^T Y^{-1}(v_2 + Z w_2) \right)$$

If I calculated correctly, this is the actual Riemann form of your torus, because it comes from a Hermitian form $Y^{-1}$ on $\mathbb C^2$ whose imaginary part resticts to the standard symplectic from on $\mathbb Z^4$.

The action of $SP_{4}(\mathbb Z)$ on this is the natural action on quadratic forms in four variables. Indeed, $v^T Y^{-1} v$ on $\mathbb C^2$ is the unique Hermitian form whose imaginary part restricts to the standard symplectic form on the lattice $\mathbb Z^4$ embedded by the map $(v_1, v_2) \mapsto v_1 + Z v_2$. The action of $SP_{4}$ on $Z$ comes from the action of $SP_4$ on this lattice plus some weird action on $\mathbb C^2$, but since the action preserves the standard symplectic form it will preserve the uniquely determined Hermitian form.

Now I think a basis of neighborhoods of the rank $k$ component of the boundary of this lattice consists of points where there are $k$ vectors $v_1,\dots,v_k$ in $\mathbb Z^4$ that are linearly independent, whose pairing under the symplectic forms with each other all vanish, and whose lengths under the quadratic form are all $<\epsilon$.

I think this because if you have a sequence of such quadratic forms with vectors $v_1,\dots,v_k$ so that the length of those vectors goes to $0$, you obtain a lattice of rank $4-2k$ with a symplectic form by modding out by $v_1,\dots,v_k$ and taking the kernel of the map to $v_1,\dots, v_k$, and you obtain a corresponding quadratic form by taking the limit of the original quadratic form, which will be well-defined as the lengths of $v_1,\dots,v_k$ goes to $0$. So a sequence of points that are in all the rank $k$ neighborhoods but not all the rank $k+1$ neighborhoods will have an accumulation point in $\mathcal A^{2-k}$. That is precisely what the Satake compactification is supposed to look like. I will try to rigorously justify this...

In fact the condition that $v_1,\dots,v_k$ have zero pairing under the symplectic form is irrelevant. As the symplectic form and quadratic form come from the same Hermitian forrm, they are related by a Cauchy-Schwartz inequality, so if $v_1,v_2$ have length at most $\epsilon$, their pairing is at most $\epsilon^2$, so because the symplectic form is integral it is $0$. So we can express the condition only in terms of tuples of short vectors in the quadratic form with matrix

$$\begin{pmatrix} Y^{-1} & Y^{-1} X \\ X Y^{-1} & XY^{-1}X + Y\end{pmatrix}$$

That's totally explicit but might not be too fun. An alternate desciption of the system of neighborhoods of the rank $2$ part which may or may not be easier is the $SP_4(\mathbb Z)$ orbit of the set of pairs $X,Y$ where the shortest vector of $Y$ has length at least $1/\epsilon$. This is the same because you can always transform your two short vectors to be the first two basis vectors.

• Awesome! Thanks a lot for this. Yeah, the quadratic form you describe is the `hodge form', and everything you said looks right. The final description is indeed what i was looking for: I don't care to describe all such Z, i just want a neighborhood in H_2 which projects down to a neighborhood in A_2. Apr 1, 2016 at 7:55